Question: Find one value of $x$ that is a solution to the equation: $(4x-1)^2=20x-5$ $x=$
Answer: We could solve for $x$ by expanding $(4x-1)^2$, combining terms that are alike, and using the quadratic formula or factoring to solve for $x$. However there is a shorter and more elegant way to approach this problem. Let's use structural features to rewrite the equation in a simpler form. Note that $20x-5=5({4x-1})$. This means that we can rewrite the equation as: $({4x-1})^2=5({4x-1})$ If we let ${p}={4x-1}$, we can see that this equation is in the form: ${p}^2=5{p}$ Let's solve this equation in terms of ${p}$ : $\begin{aligned}{p}^2&=5{p}\\\\ {p}^2-5{p}&=0\\\\ {p}({p}-5)&=0\\\\ {p}=0\ &\text{or} \ \ {p}=5 \end{aligned}$ Since ${p}={4x-1}$, let's substitute this value back into our two solutions in order to solve for $x$ : ${4x-1}=0\ \ \ \text{or} \ \ \ {4x-1}=5$ When we solve ${4x-1}=0$, we find that $x=\dfrac{1}{4}$. When we solve ${4x-1}=5$, we find that $x=\dfrac{3}{2}$. In conclusion, the two solutions of the equation $(4x-1)^2=20x-5$ are $x=\dfrac{1}{4}$ and $x=\dfrac{3}{2}$. [Is there another way to solve for x?]